Stabilized finite element solution of optimal control problems in computational fluid dynamics
Abraham, Feby V.
Doctor of Philosophy thesis
This thesis discusses the solution of optimal flow control problems, with an emphasis on solving optimal design problems involving blood as the fluid. The discretization of the governing equations of fluid flow is accomplished using stabilized finite element formulations. Although frequently and successfully applied, these methods depend on significant mesh refinement to establish strong consistency properties, when using low-order elements. We present an approach to improve the consistency properties of such methods. We develop the methodology for the numerical solution of optimal control problems using the aforementioned discretization scheme. For two possible approaches in which the optimal control problem can be discretized---optimize-then-discretize and discretize-then-optimize---we use a boundary control problem governed by the linear Oseen equations to numerically explore the influence of stabilization. We also present indicators for assessing the quality of the computed solution. We then investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the Carreau-Yasuda model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. For a steady flow scenario, we apply gradient-based optimization procedure to a benchmark problem of flow through a right-angle cannula, and to a flow through an idealized arterial graft. We present the issues involved in solving large-scale optimal design problems, and state the numerical formulations for the various approaches that could be used to solve such problems. We numerically demonstrate optimal shape design for unsteady flow in an arterial graft.
Mathematics; Mechanical engineering; Biophysics